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Unformatted text preview: Math 400
Assignment #4 (corrected) 1. Due: May 11, 2010 Find approximate numerical solutions on the interval [ 0, 2] for the differential equation
dx
= t  2 x , with initial condition t0 = 0, x0 = 1
dt
Use Euler's method, the modified Euler method, and a fourth order Runge  Kutta method,
each with step sizes h = 0.2, 0.1, and 0.05.
5
1
Show that x = x ( t ) = e −2 t + ( 2t 1) is the exact solution to this initial value problem.
4
4
For each approximate solution provide a table, a graph, and the relative error at t = 1 and t = 2. 2. Find approximate numerical solutions on the interval [ 0,1.2] for the differential equation
d2x
′
+ 4π 2 x = 0 , with initial condition t0 = 0, x0 = 1, x0 = 0
2
dt
Use Euler's method with step sizes h = 0.2, 0.1, and 0.05.
Then show that x = x ( t ) = cos 2π t is the exact solution to this initial value problem.
For each approximate solution provide a table, a graph, and the relative error at t = .5 and t = 1. 3. 4. 1
= − cos x , correct to seven
1 + e −2 x
decimal places. (Suggestion: first graph the functions to see about where the roots lie.)
Find the two smallest positive roots of the equation The system, 3x 2 − y 2 = 0 has a solution not too far from the point (1, 1). Find this root two
3 xy 2 − x 2 = 1
ways. First using the twodimensional version of Newton’s method, and second by eliminating
y, solving for x using the onedimensional Newton’s method, and substituting to find y. ...
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 Spring '11
 Staff
 Math, Numerical Analysis

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